Nuprl Lemma : blended-real-req
∀[k:ℕ+]. ∀[x,y:ℝ]. blended-real(k;x;y) = y supposing |x - y| ≤ (r1/r(k))
Proof
Definitions occuring in Statement :
blended-real: blended-real(k;x;y),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
cand: A c∧ B,
real: ℝ,
uiff: uiff(P;Q),
guard: {T},
implies: P ⇒ Q,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
int_upper: {i...},
le: A ≤ B,
so_apply: x[s],
accelerate: accelerate(k;f),
blended-real: blended-real(k;x;y),
blend-seq: blend-seq(k;x;y),
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
real-regular,
less_than_wf,
req-iff-bdd-diff,
blended-real_wf,
accelerate_wf,
regular-int-seq_wf,
nat_plus_wf,
accelerate-bdd-diff,
req_transitivity,
req_witness,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
real_wf,
eventually-equal-implies-bdd-diff,
int_upper_wf,
all_wf,
equal_wf,
less_than_transitivity1,
value-type-has-value,
int-value-type,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
int_upper_properties,
itermMultiply_wf,
intformle_wf,
int_term_value_mul_lemma,
int_formula_prop_le_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
false_wf,
not-lt-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-zero,
add-swap,
add-commutes,
zero-add,
le-add-cancel,
int_subtype_base,
equal-wf-base,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
hypothesis,
because_Cache,
independent_isectElimination,
setElimination,
rename,
dependent_functionElimination,
functionExtensionality,
applyEquality,
productElimination,
independent_functionElimination,
inrFormation,
unionElimination,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
callbyvalueReduce,
sqleReflexivity,
multiplyEquality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
divideEquality,
addEquality,
addLevel
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[x,y:\mBbbR{}]. blended-real(k;x;y) = y supposing |x - y| \mleq{} (r1/r(k))
Date html generated:
2017_10_03-AM-10_08_56
Last ObjectModification:
2017_07_05-PM-04_27_51
Theory : reals
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